$\begingroup$For the first problem : can you define what a subgraph of $G$ is ? If it is just a subset of $E$, with corresponding vertices, then just take the connected component with maximum weight ? I don't think the planarity changes anything here. For the second problem : If $k$ is fixed (i.e. not part of the input), then the problem is polynomial : check the ${n \choose k}$ possible subsets of $V$ with size $k$.$\endgroup$
– GBatJun 24 at 7:48

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$\begingroup$Wait, I got it for the first problem, I didn't realize that the weights could be negative numbers.$\endgroup$
– GBatJun 24 at 7:54